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Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

I used the explicit formula for prime counting function $\pi(x)$ and integrated the given prime function with measure as $\pi(x)$. But i couldn't deduce an exact value.

Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

I used the explicit formula for prime counting function $\pi(x)$ and integrated the given prime function with measure as $\pi(x)$. But i couldn't deduce an exact value.

(Answer I'm expecting is zero)

Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

I used the explicit formula for prime counting function $\pi(x)$ and integrated the given prime function with measure as $\pi(x)$. But i couldn't deduce an exact value. (If it is of any help, the answer I'm expecting is $(\gamma-3)$)

Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

I used the explicit formula for prime counting function $\pi(x)$ and integrated the given prime function with measure as $\pi(x)$. But i couldn't deduce an exact value.

Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

Tried doing so using proof of Merten's theorem but failed!

Consider the following sum of function of primes:

$$-\sum_{p}\ln\left( 1 - \frac{1}{(ep)^{1/2}} \right){\ln(p)}$$

Here $p$ runs through all primes and $e$ is Euler's constant.

We can see that the sum diverges.

I have following questions :

Is possible to regularize this sum ? If yes, how to do so?

Any advice about going around this is welcome. Any insights in such type of problems is/are also welcome.

Related: A question on assigning finite values to divergent sums involving expression of primes

On modified Euler product

I used the explicit formula for prime counting function $\pi(x)$ and integrated the given prime function with measure as $\pi(x)$. But i couldn't deduce an exact value. (If it is of any help, the answer I'm expecting is $(\gamma-3)$)